Time & Location: All talks will be in Gibson Hall 126-A on Thursdays at 3:30 pm unless otherwise noted. In order to have time to talk informally with the speakers, we will schedule a time we call “Tea with the speaker” that everyone is welcome to join.
Organizer: Tommaso Buvoli and SamuelPunshon-Smith
Thursday, September 25
Colloquium
Topic: Riding the neural waves: Mathematics of nonlocally coupled waves in active media
Speaker: Bard Ermentrout - University of Pittsburgh (Host: Lisa Fauci)
Abstract: Recent improvements in technology have enabled neuroscientists to simultaneously record activity of many neurons at high spatial and temporal resolution. This has allowed them to discover that activity is organized into a variety of spatial patterns such as plane waves, bullseyes, and rotating waves. In this talk, I want to distinguish two different classes or wave-like activity: (1) evoked waves or "trigger waves", and (2) phase waves. In the former, the onset of activity in one area requires prior activity in a neighboring area, while in the latter, the apparent wave motion is a consequence of timing differences between areas. I will present some recent results on the role of inhibition in controlling the propagation and stability of trigger waves. Next, I will consider coupled phase equations that describe spatio-temporal activity in intrinsically oscillatory media. I will describe recent work where we are able to extract hidden waves from human cortical recordings. Finally, I will present some work showing how ongoing phase waves can promote the propagation of trigger waves in an anisotropic manner.
Location: Gibson Hall 126-A
Time: 3:30 pm
Thursday, October 16
Colloquium
Topic: What are solitons?
Speaker: Deniz Bilman - Affiliation: University of Cincinnati (Host): Ken McLaughlin
Abstract: This is an introductory talk on the story of mathematical research on waves, aimed at graduate students from all areas and undergraduate students. This will serve as both a colloquium talk and a precursor to a self-contained mini course that will be given by the speaker in our “Integrability and Beyond!!!” Seminar, starting on October 20.
Location: Gibson Hall 126-A
Time: 3:30 pm
Thursday, October 23
Colloquium
Topic: A summation formula for Hurwitz class numbers
Speaker: Kalani Thalagoda - Tulane University
Abstract: The Hurwitz class numbers, $H(n)$, count SL$_2(\mathbb{Z})$-classes of binary quadratic forms inversely weighted by stabilizer size. They are famously connected to the sum of three squares problem and to class numbers of imaginary quadratic fields. The work of Zagier in 1975 showed that their generating functions are related to a weight $3/2$ Harmonic Maass form. In this talk, I will discuss a summation formula we obtained for the Hurwitz class numbers generating function. This is joint work with Olivia Beckwith, Nicholas Diamantis, Rajat Gupta, and Larry Rolen
Location: Gibson Hall 414
Time: 3:30 pm
Thursday, October 30
Colloquium
Topic: Quantitative topology
Speaker: Fedya Manin - University of Toronto (Host: Rafal Komendarczyk)
Abstract: Traditionally, algebraic and geometric topology focuses on classifying geometric objects (e.g. manifolds or knots) up to some equivalence relation. Typically an equivalence between two objects is realized by some third object (such as a mapping or deformation). Quantitative topology asks: how "obvious" is the equivalence relation? That is, if one takes two relatively simple equivalent objects, could it be that the simplest equivalence between them is extremely geometrically complicated? The answers turn out to vary greatly and rely on tools not just from algebraic topology and differential geometry, but also theoretical computer science and harmonic analysis among other areas. I will give an overview of a few of the things we have learned since the 1970s when Gromov initiated this program, and especially in the last decade.
Location: Gibson Hall 126-A
Time: 3:30 pm
Thursday, November 13
Colloquium
Topic: Celestial Mechanics meets Tropical Geometry
Speaker: Anton Leykin - Affiliation Georgia Tech (Host): Kalina Mincheva and Daniel Bernstein
Abstract: Given initial positions and velocities of $n$ celestial bodies, with only gravitation force in play, can you describe their trajectories?
This analytic question---the so-called $n$-body problem---prompts another problem: describe configurations of the celestial bodies that are in relative equilibrium, that is, the mutual distances don't change over time. The latter question can be set up purely algebraically, as a system of multivariate polynomial equations depending on masses of the bodies as parameters.
In a joint work with Anders Jensen, we investigate the following conjecture: for fixed $n$, up to natural symmetries, the set of planar relative equilibria for $n$ bodies with positive masses is finite. Although this statement may appear deceptively simple, it is the sixth problem on Smale’s list of problems for the 21st century and is fully resolved only for up to four bodies. We provide a computer-assisted proof for $n=5$ in the case of generic masses. The human component of the argument draws on several elementary ideas from tropical geometry, and our approach places the case $n=6$ within reach.
This talk is aimed at a general mathematical audience, including students. I will give an overview of basic celestial mechanics, basic tropical geometry, and basic rigidity theory. Examples for all of these will be in the plane.
Location: Gibson Hall 126-A
Time: 3:30 pm
